On the upper bound side, Korhonen and Rybicki [1]} showed that \(p\) -cycle detection for any constant \(p\)  can be done in a linear in \(n\)  number of rounds. For odd values of \(p\) , this is optimal due to the above lower bound. They also showed that this can be done faster for degenerate graphs. Fischer, Gonen, Kuhn, and Oshman [2]} showed how to detect \(2p\) -cycles within \(O(n^{1-1/p(p-1)})\)  rounds, for \(p\ge 2\) . This was subsequently improved by Eden, Fiat, Fischer, Kuhn, and Oshman [3]}, who showed how to detect \(2p\) -cycles in \(\tilde{O}_p(n^{1-2/(p^2-p+2)})\)  rounds for odd \(p \ge 3\) , and in \(\tilde{O}_p(n^{1-2/(p^2-2p+4)})\)  rounds for even \(p \ge 4\) . They also show that as opposed to the case of 4-cliques, the listing variant of 4-cycles is harder than its detection counterpart, requiring \(\tilde{\Omega }(n)\)  rounds. For \(p=3,4,5\) , Censor-Hillel, Fischer, Gonen, Oshman, Le Gall, and Leitersdorf [4]} then showed improved algorithms (randomized) for detecting \(2p\) -cycles, which completes within \(\tilde{O}(n^{1-1/p})\)  rounds. Still, the upper bounds here are above the respective \(\tilde{\Omega }(n^{1/2})\)  lower bound.
